Local and global estimates for hyperbolic equations in Besov–Lipschitz and Triebel–Lizorkin spaces

Abstract

We establish optimal local and global Besov–Lipschitz and Triebel–Lizorkin estimates for the solutions to linear hyperbolic partial differential equations. These estimates are based on local and global estimates for Fourier integral operators that span all possible scales (and in particular both Banach and quasi-Banach scales) of Besov–Lipschitz spaces $B_{p,q}^s\left({ℝ}^n\right)$ and certain Banach and quasi-Banach scales of Triebel–Lizorkin spaces $F_{p,q}^s\left({ℝ}^n\right)$.